Logic Statements and Definitions Quiz
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Questions and Answers

What is a statement?

  • A question
  • An exclamatory sentence
  • A declarative sentence that is either true or false (correct)
  • A command

What is a tautology?

  • A statement that is always false
  • A statement that is always true (correct)
  • A statement with a single truth value
  • A statement that depends on conditions

What is a contradiction?

  • A statement that cannot be evaluated
  • A statement that is always false (correct)
  • A statement that is true in some contexts
  • A statement that is sometimes true

What is a contingency?

<p>A statement that is not a tautology and not a contradiction (B)</p> Signup and view all the answers

What defines a predicate?

<p>A declarative whose T/F value depends on one or more variables (B)</p> Signup and view all the answers

What is a counterexample?

<p>An example that demonstrates a statement is false (C)</p> Signup and view all the answers

An integer n is even if there exists an integer k such that n = 2k.

<p>True (A)</p> Signup and view all the answers

An integer n is odd if there exists an integer k such that n = 2k.

<p>False (B)</p> Signup and view all the answers

What does the notation a|b represent?

<p>a divides b if there exists an integer k such that a*k=b</p> Signup and view all the answers

What is the definition of a rational number?

<p>If there exist a, b in the integers, b ≠ 0, such that x = a/b (A)</p> Signup and view all the answers

What is a set?

<p>A collection of objects (elements of the set)</p> Signup and view all the answers

Define a subset.

<p>A is a subset of S if every element of A is also an element of S</p> Signup and view all the answers

The empty set contains at least one element.

<p>False (B)</p> Signup and view all the answers

What defines a function?

<p>A well-defined rule that assigns a single element of Y to each element of X (D)</p> Signup and view all the answers

What is the domain of a function?

<p>X is the domain of f</p> Signup and view all the answers

What is the codomain of a function?

<p>Y is the codomain of f</p> Signup and view all the answers

A function is one-to-one (injective) if for all a, b in X, if f(a) = f(b) then a = b.

<p>True (A)</p> Signup and view all the answers

A function is onto (surjective) if for all y in Y there exists x in X such that f(x) = y.

<p>True (A)</p> Signup and view all the answers

A function is one-to-one correspondence (bijective) if it is either one-to-one or onto.

<p>False (B)</p> Signup and view all the answers

What is the inverse of a function f?

<p>If f:X→Y is one-to-one, then we can define a function f⁻¹: Range(f)→X.</p> Signup and view all the answers

What is a relation in the context of set theory?

<p>Given a set S, a relation R is a subset SxS.</p> Signup and view all the answers

An equivalence relation must be reflexive, symmetric, and transitive.

<p>True (A)</p> Signup and view all the answers

What does the Principle of Mathematical Induction entail?

<p>Show P(1) is true and P(k) → P(k+1) for k&gt;1.</p> Signup and view all the answers

What is the well-ordered principle?

<p>Every nonempty subset of the natural numbers has a least element.</p> Signup and view all the answers

What does the Pigeonhole Principle state?

<p>If n pigeons and r holes, and n &gt; r, then some pigeons must share a hole.</p> Signup and view all the answers

What does n-to-one mean in function terminology?

<p>If |f⁻¹({y})|=n, then f is n-to-one for every y in f(x).</p> Signup and view all the answers

Flashcards

Statement

A declarative sentence that can be classified as true or false.

Tautology

A statement that is unconditionally true, regardless of the truth values of its components.

Contradiction

A statement that is inherently false in all scenarios.

Contingency

A statement that is neither a tautology nor a contradiction; its truth value depends on the truth values of its components.

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Predicate

A declarative sentence whose truth value is dependent on one or more variables.

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Counterexample

An example that demonstrates the falsity of a statement by providing a scenario where the statement doesn't hold true.

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Even Number

An integer n for which there exists another integer k such that n = 2k.

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Odd Number

An integer n for which there exists another integer k such that n = 2k + 1.

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Divides

An expression a | b denotes that there exists an integer k so that a * k = b.

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Rational Number

An expression x is rational if it can be expressed as a/b where a and b are integers, and b ≠ 0.

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Set

A collection of distinct objects, referred to as elements.

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Subset

Set A is a subset of set S if all elements of A are contained within S.

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Empty Set (∅)

A set that contains no elements.

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Function

A well-defined rule assigning each element in set X to a single element in set Y.

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Domain

The set X from which the function f is defined.

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Codomain

The set Y, indicating the possible outputs of function f.

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One-to-one (Injective)

A function where distinct elements in X map to distinct elements in Y.

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Onto (Surjective)

A function where every element in Y corresponds to at least one element in X.

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One-to-one Correspondence (Bijective)

A function that is both injective and surjective.

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Inverse of f

For an injective function f from X to Y, f⁻¹ maps elements from the range back to X.

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Relation

A subset of the Cartesian product S x S defining relationships between elements.

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Equivalence Relation

A relation that satisfies reflexivity, symmetry, and transitivity.

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Principle of Mathematical Induction

To prove a statement P(n) for all natural numbers n, verify the base case and that P(k) implies P(k+1).

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Well Ordered Principle

Every nonempty subset of natural numbers has a least element.

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Pigeonhole Principle

If there are more pigeons than holes, at least one hole must contain more than one pigeon.

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n-to-one Function

In finite sets X and Y, a function f is n-to-one if each output in Y has exactly n pre-images in X.

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Study Notes

Logic and Statements

  • Statement: A declarative that can be classified as true or false.
  • Tautology: A statement that is unconditionally true, irrespective of the truth values of its components.
  • Contradiction: A statement that is inherently false in all scenarios.
  • Contingency: A statement that is neither a tautology nor a contradiction.

Mathematical Concepts

  • Predicate: A declarative sentence whose truth value is dependent on one or more variables.
  • Counterexample: Illustrates the falsity of a statement by providing an example where the statement does not hold.

Number Properties

  • Even Number: Defined as an integer n for which there exists an integer k such that n = 2k.
  • Odd Number: Defined as an integer n for which there exists an integer k such that n = 2k + 1.

Divisibility

  • Divides: An expression a|b denotes that there exists an integer k so that a * k = b.
  • Rational Number: An expression x is rational if it can be expressed as a/b where a, b are integers and b ≠ 0.

Set Theory

  • Set: A collection of distinct objects, referred to as elements.
  • Subset: Set A is a subset of set S if all elements of A are contained within S.
  • Empty Set (∅): A set that contains no elements.

Functions

  • Function: A well-defined rule assigning each element in set X to a single element in set Y.
  • Domain: The set X from which the function f is defined.
  • Codomain: The set Y, indicating the possible outputs of function f.
  • One-to-one (Injective): A function where distinct elements in X map to distinct elements in Y.
  • Onto (Surjective): A function where every element in Y corresponds to at least one element in X.
  • One-to-one Correspondence (Bijective): A function that is both injective and surjective.
  • Inverse of f: For an injective function f from X to Y, f⁻¹ maps elements from the range back to X.

Relations

  • Relation: A subset of the Cartesian product S x S defining relationships between elements.
  • Equivalence Relation: A relation that satisfies reflexivity, symmetry, and transitivity.

Mathematical Principles

  • Principle of Mathematical Induction: To prove a statement P(n) for all natural numbers n, verify the base case and that P(k) implies P(k+1).
  • Well Ordered Principle: Every nonempty subset of natural numbers has a least element.
  • Pigeonhole Principle: If there are more pigeons than holes, at least one hole must contain more than one pigeon.

Cardinality

  • n-to-one Function: In finite sets X and Y, a function f is n-to-one if each output in Y has exactly n pre-images in X.

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Description

Test your understanding of key concepts in logic with this quiz. You'll encounter terms such as statement, tautology, contradiction, and contingency. Each term is defined, and your task is to associate the correct definitions with the appropriate concepts.

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